Abstract
We establish small-time asymptotic expansions for heat kernels of hypoelliptic Hormander operators in a neighborhood of the diagonal, generalizing former results obtained in particular by Metivier and by Ben Arous. The coefficients of our expansions are identified in terms of the nilpotentization of the underlying sub-Riemannian structure. Our approach is purely analytic and relies in particular on local and global subelliptic estimates as well as on the local nature of small-time asymptotics of heat kernels. The fact that our expansions are valid not only along the diagonal but in an asymptotic neighborhood of the diagonal is the main novelty, useful in view of deriving Weyl laws for subelliptic Laplacians. In turn, we establish a number of other results on hypoelliptic heat kernels that are interesting in themselves, such as Kac's principle of not feeling the boundary, asymptotic results for singular perturbations of hypoelliptic operators, global smoothing properties for selfadjoint heat semigroups.
Highlights
Introduction and main resultLet n and m be nontrivial integers
The second main novelty is that we identify the functions in the small-time asymptotic expansion of the heat kernel in terms of the so-called nilpotentization of the sub-Riemannian structure (M, D, g) where g is a metric on D defined thanks to the vector fields X1, . . . , Xm
We establish two general results for parameter-dependent hypoelliptic heat kernels: – In Section 3.2.1, we prove that the small time asymptotics of hypoelliptic heat kernels is purely local: this reflects the famous Kac’s principle of not feeling the boundary
Summary
Let n and m be nontrivial integers. Let M be a smooth connected manifold of dimension n. The first main novelty here is that our expansion is valid, along the diagonal, but in an asymptotic neighborhood of the diagonal This fact is instrumental in view of deriving local and microlocal Weyl laws for general subelliptic Laplacians, which will be done in the forthcoming paper [CdVHT]. The second main novelty is that we identify the functions in the small-time asymptotic expansion of the heat kernel in terms of the so-called nilpotentization of the sub-Riemannian structure (M, D, g) where g is a metric on D defined thanks to the vector fields X1, . We recall that the set C∞(Rn) of smooth functions on Rn is a Montel space, i.e., a Fréchet space enjoying the Heine–Borel property (meaning that closed bounded subsets are compact), for the topology defined by the seminorms pi, j(f ) = max{|∂αf (x)| | |x| i, |α| j}, (i, j) ∈ N2. The integral of an integrable function f on M with respect to the smooth measure μ is denoted by M f (q) dμ(q)
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